Does Monte Carlo tree search qualify as machine learning?
Monte Carlo Tree Search is not usually thought of as a machine learning technique, but as a search technique. There are parallels (MCTS does try to learn general patterns from data, in a sense, but the patterns are not very general), but really MCTS is not a suitable algorithm for most learning problems.

AlphaZero was a combination of several algorithms. One was MCTS, but MCTS needs a function to tell it how good different states of the game might be (or else, it needs to simulate entire games). One way to handle this function in a game like chess or Go is to approximate it by training a neural network, which is what the Deep Mind researchers did. This is the learning component of AlphaZero.